Proof of Theorem BCS

Definition CSM describes the column space as the span of the set of columns of $A$. Theorem BS tells us that we can reduce the set of vectors used in a span. If we apply Theorem BS to $\csp{A}$, we would collect the columns of $A$ into a matrix (which would just be $A$ again) and bring the matrix to reduced row-echelon form, which is the matrix $B$ in the statement of the theorem. In this case, the conclusions of Theorem BS applied to $A$, $B$ and $\csp{A}$ are exactly the conclusions we desire.